Oscillation criteria for self-adjoint differential systems

Author:
William T. Reid

Journal:
Trans. Amer. Math. Soc. **101** (1961), 91-106

MSC:
Primary 34.30

DOI:
https://doi.org/10.1090/S0002-9947-1961-0133518-X

MathSciNet review:
0133518

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References | Similar Articles | Additional Information

**[1]**N. I. Achieser and I. M. Glasmann,*Theorie der linearen Operatoren im Hilbert-Raum*, Akademie-Verlag, Berlin, 1954 (German). MR**0066560****[2]**J. H. Barrett,*Two-point boundary value problems and comparison theorems for fourth-order self-adjoint differential equations and second-order matrix differential equations*, Technical Summary Report #150, April, 1960, Mathematics Research Center, U. S. Army.**[3]**G. D. Birkhoff and M. R. Hestenes,*Natural isoperimetric conditions in the calculus of variations*, Duke Math. J.**1**(1935), no. 2, 198–286. MR**1545876**, https://doi.org/10.1215/S0012-7094-35-00118-1**[4]**Gilbert A. Bliss,*Lectures on the Calculus of Variations*, University of Chicago Press, Chicago, Ill., 1946. MR**0017881****[5]**Maxime Bôcher,*Applications and generalizations of the conception of adjoint systems*, Trans. Amer. Math. Soc.**14**(1913), no. 4, 403–420. MR**1500954**, https://doi.org/10.1090/S0002-9947-1913-1500954-6**[6]**Magnus R. Hestenes,*Applications of the theory of quadratic forms in Hilbert space to the calculus of variations*, Pacific J. Math.**1**(1951), 525–581. MR**0046590****[7]**Henry Howard,*Oscillation criteria for fourth-order linear differential equations.*, Trans. Amer. Math. Soc.**96**(1960), 296–311. MR**0117379**, https://doi.org/10.1090/S0002-9947-1960-0117379-X**[8]**K. S. Hu,*The problem of Bolza and its accessory boundary value problem*(Dissertation, University of Chicago, 1932), Contributions to the Calculus of Variations, University of Chicago Press, 1931-1932, pp. 361-443.**[9]**Walter Leighton,*The detection of the oscillation of solutions of a second order linear differential equation*, Duke Math. J.**17**(1950), 57–61. MR**0032065****[10]**Walter Leighton and Zeev Nehari,*On the oscillation of solutions of self-adjoint linear differential equations of the fourth order*, Trans. Amer. Math. Soc.**89**(1958), 325–377. MR**0102639**, https://doi.org/10.1090/S0002-9947-1958-0102639-X**[11]**Marston Morse,*Sufficient conditions in the problem of Lagrange with fixed end points*, Ann. of Math. (2)**32**(1931), no. 3, 567–577. MR**1503017**, https://doi.org/10.2307/1968252**[12]**Marston Morse,*Sufficient Conditions in the Problem of Lagrange with Variable End Conditions*, Amer. J. Math.**53**(1931), no. 3, 517–546. MR**1507924**, https://doi.org/10.2307/2371163**[13]**-,*The calculus of variations in the large*, Amer. Math. Soc. Colloquium Publications, vol.**18**, 1934.**[14]**Zeev Nehari,*Oscillation criteria for second-order linear differential equations*, Trans. Amer. Math. Soc.**85**(1957), 428–445. MR**0087816**, https://doi.org/10.1090/S0002-9947-1957-0087816-8**[15]**William T. Reid,*A Boundary Value Problem Associated with the Calculus of Variations*, Amer. J. Math.**54**(1932), no. 4, 769–790. MR**1506937**, https://doi.org/10.2307/2371102**[16]**William T. Reid,*An Integro-Differential Boundary Value Problem*, Amer. J. Math.**60**(1938), no. 2, 257–292. MR**1507311**, https://doi.org/10.2307/2371292**[17]**William T. Reid,*A matrix differential equation of Riccati type*, Amer. J. Math.**68**(1946), 237–246. MR**0015610**, https://doi.org/10.2307/2371835**[18]**William T. Reid,*Oscillation criteria for linear differential systems with complex coefficients*, Pacific J. Math.**6**(1956), 733–751. MR**0084655****[19]**William T. Reid,*Adjoint linear differential operators*, Trans. Amer. Math. Soc.**85**(1957), 446–461. MR**0088625**, https://doi.org/10.1090/S0002-9947-1957-0088625-6**[20]**William T. Reid,*Principal solutions of non-oscillatory self-adjoint linear differential systems*, Pacific J. Math.**8**(1958), 147–169. MR**0098220****[21]**H. M. Sternberg and R. L. Sternberg,*A two-point boundary problem for ordinary self-adjoint differential equations of fourth order*, Canadian J. Math.**6**(1954), 416–419. MR**0061738**

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DOI:
https://doi.org/10.1090/S0002-9947-1961-0133518-X

Article copyright:
© Copyright 1961
American Mathematical Society